On differentiable vectors for representations of infinite dimensional Lie groups

In this paper we develop two types of tools to deal with differentiability properties of vectors in continuous representations pi : G -> GL(V) of an infinite dimensional Lie group G on a locally convex space V. The first class of results concerns the space V-infinity of smooth vectors. If G is a Banach-Lie group, we define a topology on the space V-infinity of smooth vectors for which the action of G on this space is smooth. If V is a Banach space, then V-infinity is a Frechet space. This applies in particular to C*-dynamical systems (A, G, alpha), where G is a Banach-Lie group. For unitary representations we show that a vector v is smooth if the corresponding positive definite function <pi(g)v, v > is smooth. The second class of results concerns criteria for C-k-vectors in terms of operators of the derived representation for a Banach Lie group G acting on a Banach space V. In particular, we provide for each k is an element of N examples of continuous unitary representations for which the space of Ck+1-vectors is trivial and the space of C-k-vectors is dense. (C) 2010 Elsevier Inc. All rights reserved.